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Global attractivity of periodic solutions in a higher order difference equation
Applied Mathematics Letters 26 (2013) 578–583
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Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
Global attractivity of periodic solutions in a higher order difference equation Chuanxi Qian Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762, USA
article
info
Article history: Received 2 November 2012 Received in revised form 3 December 2012 Accepted 4 December 2012 Keywords: Higher order difference equation Periodic solution Global attractivity Population model
abstract Consider the following higher order difference equation with periodic coefficients: xn+1 = an xn + F (n, xn−k ),
n = 0, 1, . . . ,
where {an } is a periodic sequence in (0, 1] with period p and an ̸≡ 1, F (n, x) : {0, 1, . . .} × [0, ∞) → (0, ∞) is a continuous function in x and a periodic function in n with period p, and k is a nonnegative integer. We obtain a sufficient condition such that every positive solution of the equation converges to a positive periodic solution. Applications to some difference equations derived from mathematical biology are also given. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction Our aim in this work is to study the global attractivity of periodic solutions of the following higher order nonlinear difference equation: xn+1 = an xn + F (n, xn−k ),
n = 0, 1, . . . ,
(1.1)
where {an } is a periodic sequence in (0, 1] with period p and an ̸≡ 1, F (n, x) : {0, 1, . . .} × [0, ∞) → (0, ∞) is a continuous function in x and a periodic function in n with period p, and k is a nonnegative integer. By a solution of Eq. (1.1), we mean a sequence {xn } which is defined for n ≥ −k and which satisfies Eq. (1.1) for n ≥ 0. If we let x−k , x−k+1 , . . . , x0
(1.2)
be k + 1 given nonnegative numbers with x0 > 0, then Eq. (1.1) has a unique positive solution with initial condition (1.2). When an ≡ α and bn ≡ β are positive constants and F (n, x) = β f (x), Eq. (1.1) reduces to the form xn+1 = α xn + β f (xn−k )
(1.3)
which includes several discrete models derived from mathematical biology. The global attractivity of positive solutions of Eq. (1.3) and applications have been studied by numerous authors; see, for example, [1–5] and references cited therein. Besides their theoretical interest, difference equations with periodic coefficients are important in mathematical biology. A model of this type could be used to mimic a population’s response to seasonal fluctuations in its environment or a population with several discrete life-cycle stages. The existence of one or more periodic solutions for Eq. (1.1) and some related forms has been studied by several authors and numerous results have been obtained on this topic; see, for example, [6–9] and references cited therein. However, studies of global attractivity of periodic solutions are scarce. In this work, we obtain a sufficient condition for every positive solution of Eq. (1.1) to converge to a positive periodic solution. Applications to some discrete population models are also given.
E-mail address: [email protected]. 0893-9659/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2012.12.005
C. Qian / Applied Mathematics Letters 26 (2013) 578–583
579
2. The main results The following theorem provides a sufficient condition for every positive solution of Eq. (1.1) to converge to a positive n periodic solution. For the sake of convenience, we adopt the notation i=m ai = 1 whenever m > n in the discussion. Theorem 1. Assume that F (n, x) is nonincreasing in x, and L-Lipschitz for each 0 ≤ n ≤ p − 1, that is, there are nonnegative constants Ln such that
|F (n, x) − F (n, y)| ≤ Ln |x − y|,
n = 0, 1, . . . , p − 1.
(2.1)
Suppose also that n +k
n+k
Lj
j=n
ai < 1,
n = 0, 1, . . . , p − 1.
(2.2)
i=j+1
Then Eq. (1.1) has a unique positive periodic solution {˜xn } with period p and every positive solution {xn } of Eq. (1.1) satisfies lim (xn − x˜ n ) = 0.
(2.3)
n→∞
Proof. Since F (n, x) > 0, F (n, x) is periodic in n and nonincreasing in x, from the known results (see, for example, [6–9]), it is easy to see that Eq. (1.1) has a positive periodic solution {˜xn } with period p. Clearly, if we can show that every positive solution of Eq. (1.1) converges to {˜xn }, then {˜xn } is the unique positive periodic solution. To this end, let yn = xn − x˜ n . Then {yn } satisfies yn+1 = an yn − (˜xn+1 − an x˜ n ) + F (n, yn−k + x˜ n−k ).
(2.4)
Since {˜xn } is a solution of Eq. (1.1), x˜ n+1 − an x˜ n = F (n, x˜ n−k ). Hence, it follows that yn+1 = an yn + (F (n, yn−k + x˜ n−k ) − F (n, x˜ n−k )).
(2.5)
First, assume that {xn } does not oscillate about {˜xn }. Hence, {yn } is either eventually positive or eventually negative. Let us assume that {yn } is eventually positive. The proof for the case where {yn } is eventually negative is similar and will be omitted. Hence there is a positive integer n0 such that yn > 0, n ≥ n0 . Then on noting that F is nonincreasing in x, from (2.5) we see that yn+1 ≤ an yn , n ≥ n0 + k and so it follows that
n −1
yn ≤
yn0 +k ,
ai
n > n0 + k.
i=n0 +k
On noting that an ∈ (0, 1], an is periodic and an ̸≡ 1, we see that i=n0 +k ai → 0 as n → ∞. Hence yn → 0 as n → ∞ and so (2.3) holds. Next, assume that {xn } oscillates about {˜xn } and so {yn } oscillates about zero. Then there is an increasing sequence {nr } of positive integers such that yn1 ≤ 0 and for l = 1, 2, . . .,
n−1
yn > 0
for n2l−1 < n ≤ n2l
yn ≤ 0
for n2l < n ≤ n2l+1 .
and
Observe that (2.5) yields yn+1 n
−
yn
ai
i=0
=
n−1
ai
1 n
(F (n, yn−k + x˜ n−k ) − F (n, x˜ n−k )). ai
i=0
i=0
Summing from n1 to n − 1 where n1 < n ≤ n2 , we see that yn n−1
i=0
ai
−
yn1
=
n 1 −1
i=0
ai
n−1 j=n1
1 j i=0
(F (j, yj−k + x˜ j−k ) − F (j, x˜ j−k )) ai
(2.6)
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C. Qian / Applied Mathematics Letters 26 (2013) 578–583
and so
n −1 n−1 yn1 1 yn = ai + ( F (j, yj−k + x˜ j−k ) − F (j, x˜ j−k )) . j n 1 −1 j =n 1 i =0 ai ai i=0
i=0
Since yn1 ≤ 0, it follows that when n1 < n ≤ n2 , yn ≤
n−1
n−1
ai
1 j
j=n1
i=0
(F (j, yj−k + x˜ j−k ) − F (j, x˜ j−k )) ai
i=0
=
n −1
j=n1
where that
n −1
(F (j, yj−k + x˜ j−k ) − F (j, x˜ j−k ))
ai
(2.7)
i=j+1
ai = 1. Since F (n, x) is periodic in n with period p and (2.1) holds, we may define a periodic sequence {Ln } such
n−1 i=n
Ln+p = Ln ,
n = 0, 1, . . . ,
and
|F (n, x) − F (n, y)| ≤ Ln |x − y|,
n = 0, 1, . . . .
Then it follows from (2.7) that when n1 < n ≤ n2 , yn ≤
≤
n −1
n −1
j=n1
i=j+1
n −1
|F (j, yj−k + x˜ j−k ) − F (j, x˜ j−k )|
ai
n −1
Lj
j=n1
|yj−k |.
ai
(2.8)
i=j+1
On noting (2.2) and the periodic property of {an } and {Ln }, we see that there is a positive constant c < 1 such that n+k
n +k
Lj
j =n
ai ≤ c ,
n = 0, 1, . . . .
(2.9)
i=j+1
We claim that yn ≤ c
max n1 −k≤s≤n1
n1 < n ≤ n2 .
|ys |,
(2.10)
To this end, consider the two cases n2 ≤ n1 + k + 1 and n2 > n1 + k + 1. When n2 ≤ n1 + k + 1, then for any n1 < n ≤ n2 , n − k − 1 ≤ n1 and so (2.8) yields yn ≤
n −1
Lj
j=n1
≤
n −1
ai
i=j+1
n−1
Lj
j=n−k−1
n−1
max n1 −k≤s≤n1
|ys |
ai
i =j +1
max n1 −k≤s≤n1
|ys |.
(2.11)
Then on noting (2.9), we see that (2.10) holds. Next, consider the case where n2 > n1 + k + 1. When n1 < n ≤ n1 + k + 1, as we have shown above, (2.10) holds; when n1 + k + 1 < n ≤ n2 , on noting that yn−k−1 > 0 and F is nonincreasing in x, (2.5) yields yn = an−1 yn−1 + (F (n − 1, yn−k−1 + x˜ n−k−1 ) − F (n − 1, x˜ n−k−1 )) ≤ an−1 yn−1 . Then on noting that an ∈ (0, 1], it follows that yn2 ≤ yn2 −1 ≤ · · · ≤ yn1 +1+k , which implies that (2.10) holds when n1 + k + 1 < n ≤ n2 . Hence for any n1 < n ≤ n2 , (2.10) holds. Then by a similar argument, we may show that yn ≥ −c
max n2 −k≤s≤n2
|ys |,
n2 < n ≤ n3 .
(2.12)
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If n2 − k > n1 , we see that when n2 − k ≤ n ≤ n2 , (2.10) holds and so
|ys | ≤ c
max n2 −k≤s≤n2
max n1 −k≤s≤n1
|ys | ≤
max n1 −k≤s≤n1
|ys |.
If n2 − k < n1 , we see that (2.10) holds when n1 < n ≤ n2 ; while when n2 − k ≤ n ≤ n1 , on noting that n1 − k < n2 − k, we see that |yn | ≤ maxn1 −k≤s≤n1 |ys |. Hence, from the above discussion, we see that maxn2 −k≤s≤n2 |ys | ≤ maxn1 −k≤s≤n1 |ys | and so it follows from (2.12) that when n2 < n ≤ n3 , yn ≥ −c
|ys |.
max n1 −k≤s≤n1
(2.13)
By combining (2.10) and (2.13), we see that
|yn | ≤ c
max n1 −k≤s≤n1
|ys | for n1 < n ≤ n3 .
Then by the method of steps, we may show that
|yn | ≤ c
max n1 −k≤s≤n1
|ys | for n > n1 .
Now, by choosing an nr2 ∈ {nr } with nr2 > n1 + k and then using a similar argument, we may show that
|yn | ≤ c 2
max n1 −k≤s≤n1
|ys | for n > nr2 .
Finally, by induction, we may show that for any positive integer m > 1,
|yn | ≤ c m
max n1 −k≤s≤n1
|ys | for n > nrm
where nrm ∈ {nr } with nrm > nrm−1 + k. Since 0 < c < 1, we see that yn → 0 as n → ∞ and then it follows that (2.3) holds. The proof is complete. The following corollary is a direct consequence of Theorem 1 with F (n, x) = bn f (x). Corollary 1. Consider the following difference equation: xn+1 = an xn + bn f (xn−k ),
n = 0, 1, . . . ,
(2.14)
where f : [0, ∞) → (0, ∞) is a continuous function, {an } is a periodic sequence in (0, 1] with period p and an ̸≡ 1, {bn } is a positive periodic sequence with period p, and k is a nonnegative integer. Suppose that f is nonincreasing and L-Lipschitz, and that L
n +k j =n
bj
n +k
ai < 1,
n = 0, 1, . . . , p − 1.
(2.15)
i=j+1
Then Eq. (2.14) has a unique positive periodic solution {˜xn } with period p and every positive solution {xn } of Eq. (2.14) converges to {˜xn }. Remark 1. A special case of Eq. (2.14) is the following linear equation: xn+1 = an xn + bn ,
n = 0, 1, . . . ,
(2.16)
where {an } is a periodic sequence in (0, 1] with period p and an ̸≡ 1 and {bn } is a positive periodic sequence with period p. Since f (x) ≡ 1, we may choose L = 0 and then it follows that (2.15) holds automatically. Hence, by Corollary 1, Eq. (2.16) has a unique positive periodic solution {˜xn } with period p and every positive solution of Eq. (2.16) converges to {˜xn }. In particular, when an ≡ a and bn ≡ b are positive constants and a < 1, Eq. (2.16) reduces to the linear equation with constant coefficients xn+1 = axn + b.
(2.17)
b The positive periodic solution {˜xn } of the equation becomes the positive equilibrium point x¯ = 1− . In fact, for this trivial a case, we know that the solutions of Eq. (2.17) are in the form
x n = an x 0 + b
1 − an 1−a
,
b and so xn → 1− as n → ∞. a
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C. Qian / Applied Mathematics Letters 26 (2013) 578–583
3. Applications In this section, we apply the results obtained in the last section to some equations derived from mathematical biology. Consider the difference equations xn+1 = an xn + bn e−σn xn−k ,
n = 0, 1, . . . ,
(3.1)
and xn+1 = an xn +
bn
γ
1 + xn−k
,
n = 0, 1, . . . ,
(3.2)
where {an } is a periodic sequence in (0, 1] with period p and an ̸≡ 1, {bn } and {σn } are positive sequences with period p, γ is a positive constant and k is a nonnegative integer. When an ≡ a, bn ≡ b and σn ≡ σ are positive constants, Eqs. (3.1) and (3.2) become xn+1 = axn + be−σ xn−k ,
n = 0, 1, . . . ,
(3.3)
and xn+1 = axn +
b γ
1 + xn−k
,
n = 0, 1, . . . ,
(3.4)
respectively. Eq. (3.3) is a discrete version of a model of the survival of red blood cells in an animal [10], and Eq. (3.4) is a discrete analogue of a model that has been used to study blood cell production [11]. The global attractivity of positive solutions of Eqs. (3.3) and (3.4) has been studied by numerous authors; see, for example, [1–5] and references cited therein. Clearly, Eq. (3.1) is in the form of Eq. (1.1) with F (n, x) = bn e−σn x . Observing dF dx
= −bn σn e−σn x ≤ 0,
x ≥ 0,
we see that F (n, x) is nonincreasing in x and | dF | ≤ bn σn for x ≥ 0 which implies that for each n, F (n, x) is L-Lipschitz with dx Ln = bn σn . Hence, we have the following conclusion from Theorem 1. Corollary 2. Assume that n +k
bj σj
j =n
n +k
ai < 1 ,
n = 0, 1, . . . , p − 1.
i=j+1
Then Eq. (3.1) has a unique positive periodic solution {˜xn } with period p and every positive solution {xn } of Eq. (3.1) converges to {˜xn }. Next, consider Eq. (3.2). It is in the form of (2.14) with f (x) = 1+1xγ . Observing f ′ (x) = −
γ x γ −1 ( 1 + xγ ) 2
and f ′′ (x) =
xγ −2 ((γ + 1)xγ − (γ − 1))
(1 + xγ )3 γ −1
1
we see that when γ = 1, |f ′ (x)| ≤ |f ′ (0)| = 1, x > 0, and when γ > 1, |f ′ (x)| attains its maximum at x∗ = ( γ +1 ) γ and
|f ′ (x∗ )| =
(γ − 1)
γ −1 γ
(γ + 1) 4γ
γ +1 γ
. γ −1
Hence, f is L-Lipschitz with L = 1 when γ = 1, and with L = that the following conclusion holds.
γ +1
(γ −1) γ (γ +1) γ 4γ
when γ > 1. Then it follows from Corollary 1
Corollary 3. Assume that either γ = 1 and n +k
bj
j =n
n +k
ai < 1,
n = 0, 1, . . . , p − 1,
i=j+1
or γ > 1 and n +k j =n
bj
n +k i=j+1
ai <
4γ
(γ − 1)
γ −1 γ
(γ + 1)
γ +1 γ
,
n = 0, 1, . . . , p − 1.
Then Eq. (3.2) has a unique positive periodic solution {˜xn } with period p and every positive solution {xn } of Eq. (3.2) converges to {˜xn }.
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
H.A. El-Morshedy, E. Liz, Convergence to equilibria in discrete population models, J. Difference Equ. Appl. 2 (2005) 117–131. J.R. Graef, C. Qian, Global stability in a nonlinear difference equation, J. Difference Equ. Appl. 5 (1999) 251–270. A.F. Ivanov, On global stability in a nonlinear discrete model, Nonlinear Anal. 23 (1994) 1383–1389. G. Karakostas, Ch.G. Philos, Y.G. Sficas, The dynamics of some discrete population models, Nonlinear Anal. 17 (1991) 1069–1084. V.L. Kocic, G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993. S.S. Cheng, G. Zhang, Positive periodic solutions for discrete population models, Nonlinear Funct. Anal. Appl. 8 (2003) 335–344. Y. Raffoul, E. Yankson, Positive periodic solutions in neutral delay difference equations, Adv. Dyn. Syst. Appl. 5 (2010) 123–130. J. Wu, Y. Liu, Two periodic solutions of neutral difference equations modelling physiological processes, Discrete Dyn. Nat. Soc. (2006) 12. Art. ID 78145. G. Zhang, S.G. Kang, S.S. Cheng, Periodic solutions for a couple pair of delay difference equations, Adv. Differential Equations 3 (2005) 215–226. M. Wazewska-Czyzewska, A. Lasota, Mathematical problems of the dynamics of the red-blood cells system, Ann. Polish Math. Soc. Ser. III Appl. Math. 17 (1988) 23–40. M.C. Mackey, L. Glass, Oscillation and chaos in physiological control systems, Science 197 (1977) 287–289.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
Global attractivity of periodic solutions in a higher order difference equation Chuanxi Qian Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762, USA
article
info
Article history: Received 2 November 2012 Received in revised form 3 December 2012 Accepted 4 December 2012 Keywords: Higher order difference equation Periodic solution Global attractivity Population model
abstract Consider the following higher order difference equation with periodic coefficients: xn+1 = an xn + F (n, xn−k ),
n = 0, 1, . . . ,
where {an } is a periodic sequence in (0, 1] with period p and an ̸≡ 1, F (n, x) : {0, 1, . . .} × [0, ∞) → (0, ∞) is a continuous function in x and a periodic function in n with period p, and k is a nonnegative integer. We obtain a sufficient condition such that every positive solution of the equation converges to a positive periodic solution. Applications to some difference equations derived from mathematical biology are also given. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction Our aim in this work is to study the global attractivity of periodic solutions of the following higher order nonlinear difference equation: xn+1 = an xn + F (n, xn−k ),
n = 0, 1, . . . ,
(1.1)
where {an } is a periodic sequence in (0, 1] with period p and an ̸≡ 1, F (n, x) : {0, 1, . . .} × [0, ∞) → (0, ∞) is a continuous function in x and a periodic function in n with period p, and k is a nonnegative integer. By a solution of Eq. (1.1), we mean a sequence {xn } which is defined for n ≥ −k and which satisfies Eq. (1.1) for n ≥ 0. If we let x−k , x−k+1 , . . . , x0
(1.2)
be k + 1 given nonnegative numbers with x0 > 0, then Eq. (1.1) has a unique positive solution with initial condition (1.2). When an ≡ α and bn ≡ β are positive constants and F (n, x) = β f (x), Eq. (1.1) reduces to the form xn+1 = α xn + β f (xn−k )
(1.3)
which includes several discrete models derived from mathematical biology. The global attractivity of positive solutions of Eq. (1.3) and applications have been studied by numerous authors; see, for example, [1–5] and references cited therein. Besides their theoretical interest, difference equations with periodic coefficients are important in mathematical biology. A model of this type could be used to mimic a population’s response to seasonal fluctuations in its environment or a population with several discrete life-cycle stages. The existence of one or more periodic solutions for Eq. (1.1) and some related forms has been studied by several authors and numerous results have been obtained on this topic; see, for example, [6–9] and references cited therein. However, studies of global attractivity of periodic solutions are scarce. In this work, we obtain a sufficient condition for every positive solution of Eq. (1.1) to converge to a positive periodic solution. Applications to some discrete population models are also given.
E-mail address: [email protected]. 0893-9659/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2012.12.005
C. Qian / Applied Mathematics Letters 26 (2013) 578–583
579
2. The main results The following theorem provides a sufficient condition for every positive solution of Eq. (1.1) to converge to a positive n periodic solution. For the sake of convenience, we adopt the notation i=m ai = 1 whenever m > n in the discussion. Theorem 1. Assume that F (n, x) is nonincreasing in x, and L-Lipschitz for each 0 ≤ n ≤ p − 1, that is, there are nonnegative constants Ln such that
|F (n, x) − F (n, y)| ≤ Ln |x − y|,
n = 0, 1, . . . , p − 1.
(2.1)
Suppose also that n +k
n+k
Lj
j=n
ai < 1,
n = 0, 1, . . . , p − 1.
(2.2)
i=j+1
Then Eq. (1.1) has a unique positive periodic solution {˜xn } with period p and every positive solution {xn } of Eq. (1.1) satisfies lim (xn − x˜ n ) = 0.
(2.3)
n→∞
Proof. Since F (n, x) > 0, F (n, x) is periodic in n and nonincreasing in x, from the known results (see, for example, [6–9]), it is easy to see that Eq. (1.1) has a positive periodic solution {˜xn } with period p. Clearly, if we can show that every positive solution of Eq. (1.1) converges to {˜xn }, then {˜xn } is the unique positive periodic solution. To this end, let yn = xn − x˜ n . Then {yn } satisfies yn+1 = an yn − (˜xn+1 − an x˜ n ) + F (n, yn−k + x˜ n−k ).
(2.4)
Since {˜xn } is a solution of Eq. (1.1), x˜ n+1 − an x˜ n = F (n, x˜ n−k ). Hence, it follows that yn+1 = an yn + (F (n, yn−k + x˜ n−k ) − F (n, x˜ n−k )).
(2.5)
First, assume that {xn } does not oscillate about {˜xn }. Hence, {yn } is either eventually positive or eventually negative. Let us assume that {yn } is eventually positive. The proof for the case where {yn } is eventually negative is similar and will be omitted. Hence there is a positive integer n0 such that yn > 0, n ≥ n0 . Then on noting that F is nonincreasing in x, from (2.5) we see that yn+1 ≤ an yn , n ≥ n0 + k and so it follows that
n −1
yn ≤
yn0 +k ,
ai
n > n0 + k.
i=n0 +k
On noting that an ∈ (0, 1], an is periodic and an ̸≡ 1, we see that i=n0 +k ai → 0 as n → ∞. Hence yn → 0 as n → ∞ and so (2.3) holds. Next, assume that {xn } oscillates about {˜xn } and so {yn } oscillates about zero. Then there is an increasing sequence {nr } of positive integers such that yn1 ≤ 0 and for l = 1, 2, . . .,
n−1
yn > 0
for n2l−1 < n ≤ n2l
yn ≤ 0
for n2l < n ≤ n2l+1 .
and
Observe that (2.5) yields yn+1 n
−
yn
ai
i=0
=
n−1
ai
1 n
(F (n, yn−k + x˜ n−k ) − F (n, x˜ n−k )). ai
i=0
i=0
Summing from n1 to n − 1 where n1 < n ≤ n2 , we see that yn n−1
i=0
ai
−
yn1
=
n 1 −1
i=0
ai
n−1 j=n1
1 j i=0
(F (j, yj−k + x˜ j−k ) − F (j, x˜ j−k )) ai
(2.6)
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C. Qian / Applied Mathematics Letters 26 (2013) 578–583
and so
n −1 n−1 yn1 1 yn = ai + ( F (j, yj−k + x˜ j−k ) − F (j, x˜ j−k )) . j n 1 −1 j =n 1 i =0 ai ai i=0
i=0
Since yn1 ≤ 0, it follows that when n1 < n ≤ n2 , yn ≤
n−1
n−1
ai
1 j
j=n1
i=0
(F (j, yj−k + x˜ j−k ) − F (j, x˜ j−k )) ai
i=0
=
n −1
j=n1
where that
n −1
(F (j, yj−k + x˜ j−k ) − F (j, x˜ j−k ))
ai
(2.7)
i=j+1
ai = 1. Since F (n, x) is periodic in n with period p and (2.1) holds, we may define a periodic sequence {Ln } such
n−1 i=n
Ln+p = Ln ,
n = 0, 1, . . . ,
and
|F (n, x) − F (n, y)| ≤ Ln |x − y|,
n = 0, 1, . . . .
Then it follows from (2.7) that when n1 < n ≤ n2 , yn ≤
≤
n −1
n −1
j=n1
i=j+1
n −1
|F (j, yj−k + x˜ j−k ) − F (j, x˜ j−k )|
ai
n −1
Lj
j=n1
|yj−k |.
ai
(2.8)
i=j+1
On noting (2.2) and the periodic property of {an } and {Ln }, we see that there is a positive constant c < 1 such that n+k
n +k
Lj
j =n
ai ≤ c ,
n = 0, 1, . . . .
(2.9)
i=j+1
We claim that yn ≤ c
max n1 −k≤s≤n1
n1 < n ≤ n2 .
|ys |,
(2.10)
To this end, consider the two cases n2 ≤ n1 + k + 1 and n2 > n1 + k + 1. When n2 ≤ n1 + k + 1, then for any n1 < n ≤ n2 , n − k − 1 ≤ n1 and so (2.8) yields yn ≤
n −1
Lj
j=n1
≤
n −1
ai
i=j+1
n−1
Lj
j=n−k−1
n−1
max n1 −k≤s≤n1
|ys |
ai
i =j +1
max n1 −k≤s≤n1
|ys |.
(2.11)
Then on noting (2.9), we see that (2.10) holds. Next, consider the case where n2 > n1 + k + 1. When n1 < n ≤ n1 + k + 1, as we have shown above, (2.10) holds; when n1 + k + 1 < n ≤ n2 , on noting that yn−k−1 > 0 and F is nonincreasing in x, (2.5) yields yn = an−1 yn−1 + (F (n − 1, yn−k−1 + x˜ n−k−1 ) − F (n − 1, x˜ n−k−1 )) ≤ an−1 yn−1 . Then on noting that an ∈ (0, 1], it follows that yn2 ≤ yn2 −1 ≤ · · · ≤ yn1 +1+k , which implies that (2.10) holds when n1 + k + 1 < n ≤ n2 . Hence for any n1 < n ≤ n2 , (2.10) holds. Then by a similar argument, we may show that yn ≥ −c
max n2 −k≤s≤n2
|ys |,
n2 < n ≤ n3 .
(2.12)
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If n2 − k > n1 , we see that when n2 − k ≤ n ≤ n2 , (2.10) holds and so
|ys | ≤ c
max n2 −k≤s≤n2
max n1 −k≤s≤n1
|ys | ≤
max n1 −k≤s≤n1
|ys |.
If n2 − k < n1 , we see that (2.10) holds when n1 < n ≤ n2 ; while when n2 − k ≤ n ≤ n1 , on noting that n1 − k < n2 − k, we see that |yn | ≤ maxn1 −k≤s≤n1 |ys |. Hence, from the above discussion, we see that maxn2 −k≤s≤n2 |ys | ≤ maxn1 −k≤s≤n1 |ys | and so it follows from (2.12) that when n2 < n ≤ n3 , yn ≥ −c
|ys |.
max n1 −k≤s≤n1
(2.13)
By combining (2.10) and (2.13), we see that
|yn | ≤ c
max n1 −k≤s≤n1
|ys | for n1 < n ≤ n3 .
Then by the method of steps, we may show that
|yn | ≤ c
max n1 −k≤s≤n1
|ys | for n > n1 .
Now, by choosing an nr2 ∈ {nr } with nr2 > n1 + k and then using a similar argument, we may show that
|yn | ≤ c 2
max n1 −k≤s≤n1
|ys | for n > nr2 .
Finally, by induction, we may show that for any positive integer m > 1,
|yn | ≤ c m
max n1 −k≤s≤n1
|ys | for n > nrm
where nrm ∈ {nr } with nrm > nrm−1 + k. Since 0 < c < 1, we see that yn → 0 as n → ∞ and then it follows that (2.3) holds. The proof is complete. The following corollary is a direct consequence of Theorem 1 with F (n, x) = bn f (x). Corollary 1. Consider the following difference equation: xn+1 = an xn + bn f (xn−k ),
n = 0, 1, . . . ,
(2.14)
where f : [0, ∞) → (0, ∞) is a continuous function, {an } is a periodic sequence in (0, 1] with period p and an ̸≡ 1, {bn } is a positive periodic sequence with period p, and k is a nonnegative integer. Suppose that f is nonincreasing and L-Lipschitz, and that L
n +k j =n
bj
n +k
ai < 1,
n = 0, 1, . . . , p − 1.
(2.15)
i=j+1
Then Eq. (2.14) has a unique positive periodic solution {˜xn } with period p and every positive solution {xn } of Eq. (2.14) converges to {˜xn }. Remark 1. A special case of Eq. (2.14) is the following linear equation: xn+1 = an xn + bn ,
n = 0, 1, . . . ,
(2.16)
where {an } is a periodic sequence in (0, 1] with period p and an ̸≡ 1 and {bn } is a positive periodic sequence with period p. Since f (x) ≡ 1, we may choose L = 0 and then it follows that (2.15) holds automatically. Hence, by Corollary 1, Eq. (2.16) has a unique positive periodic solution {˜xn } with period p and every positive solution of Eq. (2.16) converges to {˜xn }. In particular, when an ≡ a and bn ≡ b are positive constants and a < 1, Eq. (2.16) reduces to the linear equation with constant coefficients xn+1 = axn + b.
(2.17)
b The positive periodic solution {˜xn } of the equation becomes the positive equilibrium point x¯ = 1− . In fact, for this trivial a case, we know that the solutions of Eq. (2.17) are in the form
x n = an x 0 + b
1 − an 1−a
,
b and so xn → 1− as n → ∞. a
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C. Qian / Applied Mathematics Letters 26 (2013) 578–583
3. Applications In this section, we apply the results obtained in the last section to some equations derived from mathematical biology. Consider the difference equations xn+1 = an xn + bn e−σn xn−k ,
n = 0, 1, . . . ,
(3.1)
and xn+1 = an xn +
bn
γ
1 + xn−k
,
n = 0, 1, . . . ,
(3.2)
where {an } is a periodic sequence in (0, 1] with period p and an ̸≡ 1, {bn } and {σn } are positive sequences with period p, γ is a positive constant and k is a nonnegative integer. When an ≡ a, bn ≡ b and σn ≡ σ are positive constants, Eqs. (3.1) and (3.2) become xn+1 = axn + be−σ xn−k ,
n = 0, 1, . . . ,
(3.3)
and xn+1 = axn +
b γ
1 + xn−k
,
n = 0, 1, . . . ,
(3.4)
respectively. Eq. (3.3) is a discrete version of a model of the survival of red blood cells in an animal [10], and Eq. (3.4) is a discrete analogue of a model that has been used to study blood cell production [11]. The global attractivity of positive solutions of Eqs. (3.3) and (3.4) has been studied by numerous authors; see, for example, [1–5] and references cited therein. Clearly, Eq. (3.1) is in the form of Eq. (1.1) with F (n, x) = bn e−σn x . Observing dF dx
= −bn σn e−σn x ≤ 0,
x ≥ 0,
we see that F (n, x) is nonincreasing in x and | dF | ≤ bn σn for x ≥ 0 which implies that for each n, F (n, x) is L-Lipschitz with dx Ln = bn σn . Hence, we have the following conclusion from Theorem 1. Corollary 2. Assume that n +k
bj σj
j =n
n +k
ai < 1 ,
n = 0, 1, . . . , p − 1.
i=j+1
Then Eq. (3.1) has a unique positive periodic solution {˜xn } with period p and every positive solution {xn } of Eq. (3.1) converges to {˜xn }. Next, consider Eq. (3.2). It is in the form of (2.14) with f (x) = 1+1xγ . Observing f ′ (x) = −
γ x γ −1 ( 1 + xγ ) 2
and f ′′ (x) =
xγ −2 ((γ + 1)xγ − (γ − 1))
(1 + xγ )3 γ −1
1
we see that when γ = 1, |f ′ (x)| ≤ |f ′ (0)| = 1, x > 0, and when γ > 1, |f ′ (x)| attains its maximum at x∗ = ( γ +1 ) γ and
|f ′ (x∗ )| =
(γ − 1)
γ −1 γ
(γ + 1) 4γ
γ +1 γ
. γ −1
Hence, f is L-Lipschitz with L = 1 when γ = 1, and with L = that the following conclusion holds.
γ +1
(γ −1) γ (γ +1) γ 4γ
when γ > 1. Then it follows from Corollary 1
Corollary 3. Assume that either γ = 1 and n +k
bj
j =n
n +k
ai < 1,
n = 0, 1, . . . , p − 1,
i=j+1
or γ > 1 and n +k j =n
bj
n +k i=j+1
ai <
4γ
(γ − 1)
γ −1 γ
(γ + 1)
γ +1 γ
,
n = 0, 1, . . . , p − 1.
Then Eq. (3.2) has a unique positive periodic solution {˜xn } with period p and every positive solution {xn } of Eq. (3.2) converges to {˜xn }.
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